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arXiv:quant-ph/0302053

Representation theorem for obsevables on a quantum logic

Andrei Yu. Khrennikov, Oľga Nánásiová

Published Feb 7, 2003• 0 citations
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Abstract

We study a conditional state on a quantum logic using Renyi's approach (or Bayesian principle). This approach helps us to define independence of events and differently from the situation in the classical theory of probability, if an event $a$ is independent of an event $b$, then the event $b$ can be dependent on the event $a$. We will show that we can define a $s$-map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the $s$-map and conversely. By using the $s$-map we can introduce joint distribution also for noncompatible observables on a quantum logic.

DOI
10.48550/arxiv.quant-ph/0302053

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We study a conditional state on a quantum logic using Renyi's approach (or Bayesian principle). This approach helps us to define independence of events and differently from the situation in the classical theory of probability, if an event aaa is independent of an event bbb, then the event bbb can be dependent on the event aaa. We will show that we can define a sss-map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the sss-map and conversely. By using the sss-map we can introduce joint distribution also for noncompatible observables on a quantum logic.
This paper i a continuation of the first part under the same title. The author studies a joint distribution in σ\sigmaσ-finite measures for noncompatible observables of a quantum logic defined on some system of σ\sigmaσ-independent Boolean sub-σ\sigmaσ-algebras of a Boolean σ\sigmaσ-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
The notion of a joint distribution in σ\sigmaσ-finite measures of observables of a quantum logic defined on some system of σ\sigmaσ-independent Boolean sub-σ\sigmaσ-algebras of a Boolean σ\sigmaσ-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.
It means that if we put two conditions on the event A, the resulting conditional probability is independent of the order of conditions. We show that this is not true in quantum logics (OML) and in MV-algebras. The conditional measure (probability) in OML can be defined by using the so-called s-map. Let L be an OML. The map pn : L n → [0, 1] will be called an sn-map if the following conditions hold: (s1) pn(1, ..., 1) = 1; (s2) if there exist i, j, such that ai ⊥ aj, then pn(a1, ..., an) = 0; (s3) if ai ⊥ bi for some i = 1, ..., n, then pn(a1, ..., ai ∨ bi, ..., an) = pn(a1, ..., ai, ..., an) + pn(a1, ..., bi, ..., an).
The quantum mechanical no-cloning theorem for pure states is generalized and transfered to the quantum logics with a conditional probability calculus. This is, on the one hand, an extension of the classical probability calculus and, on the other hand, a mathematical generalization of the Lueders - von Neumann quantum measurement process. In the non-classical case, a very special type of conditional probability emerges, describing the probability for the transition from a past event to a future event independently of any underlying state. This probability results from the algebraic structure of the quantum logic only and is invariant under algebraic morphisms, which is used to prove the generalized no-cloning theorem in a rather abstract, though simple and basic fashion without relying on a tensor product construction as required in other approaches.
Abstract This paper reviews arguments for the claim that all probabilities arising in quantum mechanics are Bayesian degrees of belief. The starting point is the observation that there is a one-to-one map from quantum states to points on the probability simplex. Bayes’s rule remains valid in quantum theory, provided it is kept in mind that the outcomes of quantum measurements have no preassigned values. The probabilities of quantum measurement outcomes depend explicitly on a quantum prior whose general form will be discussed. Similar to the classical case, the quantum de Finetti theorem simplifies the analysis of repeated quantum trials if the prior is exchangeable. The Bayesian approach to quantum theory described here solves the problem of the ‘collapse of the wave function’. It also provides almost trivial explanations of a number of phenomena in quantum information theory, including the no-cloning theorem and quantum teleportation.